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Article

Координатное и импульсное туннелирование в одномерных квантовых системах с дискретным спектром

We consider the problem of constructing semiclassical asymptotic expansions of discrete spectrum and the corresponding stationary states of one-dimensional Schrödinger operator in the case of resonance tunneling. We consider two basic models: tunneling in an asymmetric double-well potential on a line and momentum tunneling of a particle in a potential field on a circle. For an asymmetric double-well potential we obtain the criterion of resonance tunneling, i. e. the necessary and sufficient conditions of stationary states bilocalization. We obtain explicit asymptotic formulas for the tunneling energy splitting in the case of high energy levels and for energies close to the minima of the potential. In the general case of dynamic tunneling we proposed a general method to find the asymptotic estimates for the tunneling energy splitting. In the case of the particle on a circle our method yields an asymptotic formula for the tunneling splitting, which is applicable in the case of analytical potential as well as in the case of finite smoothness. As an example, we consider the problem of momentum tunneling of the quantum pendulum.